Cold-Plasma Dispersion Relation

It is convenient to define a vector

$\displaystyle {\bf n} = \frac{c}{\omega}\,{\bf k}$ (5.40)

that points in the same direction as the wavevector, ${\bf k}$, and whose magnitude, $n$, is the refractive index (i.e., the ratio of the velocity of light in vacuum to the phase-velocity). Equation (5.9) can be rewritten

$\displaystyle {\bf M}\cdot{\bf E} =({\bf n}\cdot{\bf E})\,{\bf n} - n^2\,{\bf E}+ {\bf K}\cdot{\bf E} = {\bf0}.$ (5.41)

Without loss of generality, we can assume that the equilibrium magnetic field is directed along the $z$-axis, and that the wavevector, ${\bf k}$, lies in the $x$-$z$ plane. Let $\theta$ be the angle subtended between ${\bf k}$ and ${\bf B}_0$. The eigenmode equation (5.41) can be written

S - n^2\,\cos^2\theta, & -{\rm i}\...
...begin{array}{c} E_x\\ E_y \\ E_z \end{array}\!\right) = {\bf0}.\end{displaymath} (5.42)

The condition for a nontrivial solution is that the determinant of the square matrix be zero. With the help of the identity

$\displaystyle S^2 - D^{2} \equiv R\,L,$ (5.43)

we find that (Hazeltine and Waelbroeck 2004)

$\displaystyle {\cal M}(\omega,{\bf k}) \equiv A\,n^4- B\,n^2 + C = 0,$ (5.44)


$\displaystyle A$ $\displaystyle = S\,\sin^2\theta + P\,\cos^2\theta,$ (5.45)
$\displaystyle B$ $\displaystyle = R\,L\,\sin^2\theta + P\,S\,(1+\cos^2\theta),$ (5.46)
$\displaystyle C$ $\displaystyle = P\,R\,L.$ (5.47)

The dispersion relation (5.44) is evidently a quadratic in $n^2$, with two roots. The solution can be written

$\displaystyle n^2 = \frac{B\pm F}{2\,A},$ (5.48)


$\displaystyle F^{2} = (B^{2}-4\,A\,C) = (R\,L - P\,S)^2\,\sin^4\theta + 4\,P^{2} \,D^{2}\,\cos^2\theta.$ (5.49)

Note that $F^{2}\geq 0$. It follows that $n^2$ is always real, which implies that $n$ is either purely real, or purely imaginary. In other words, the cold-plasma dispersion relation describes waves that either propagate without evanescense, or decay without spatial oscillation. The two roots of opposite sign for $n$, corresponding to a particular root for $n^2$, simply describe waves of the same type propagating, or decaying, in opposite directions.

The dispersion relation (5.44) can also be written

$\displaystyle \tan^2\theta = -\frac{P\,(n^2-R)\,(n^2-L)}{(S\,n^2 - R\,L)\,(n^2-P)}.$ (5.50)

For the special case of wave propagation parallel to the magnetic field (i.e., $\theta=0$), the previous expression reduces to

$\displaystyle P$ $\displaystyle =0,$ (5.51)
$\displaystyle n^2$ $\displaystyle = R,$ (5.52)
$\displaystyle n^2$ $\displaystyle = L.$ (5.53)

Likewise, for the special case of propagation perpendicular to the field (i.e., $\theta=\pi/2$), Equation (5.50) yields

$\displaystyle n^2$ $\displaystyle = \frac{R\,L}{S},$ (5.54)
$\displaystyle n^2$ $\displaystyle = P.$ (5.55)